Discrete breathers (DBs) in nonlinear lattices have attracted much attention in the past decades. In this work, we focus on the formation of DBs and their induced energy localization in the nonlinear honeycomb lattice derived from graphene. The key step is to construct a reduced system (RS) with only a few degrees of freedom, which contains one central site and its three nearest neighbors. The fixed points and periodic orbits of the RS can be obtained from the Poincaré section of the dynamics. Our main finding is that the long-running DB solution of the full honeycomb system corresponds to the periodic orbit given by one of the fixed points of RS, where the central site and its nearest neighbors vibrate inversely. When the initial condition deviates from this fixed point, the local vibration is attracted to it after a short transient process. When the initial condition is assigned to other fixed points of the RS, the initial excitation energy flows to the other part of the full system quickly, resulting in a delocalized wave propagation. Another main finding is that the long-lived DB solutions emerge only when the initial excitation energy is larger than a threshold value, above which the frequency of the DB exceeds the phonon band edge. The excitation energy generally dissipates from the local region due to the interactions between the DB and phonons near the $\mathrm{\Gamma }$ point in the dispersion relation. These results provide a holistic physical picture for the DB solutions in two-dimensional nonlinear lattices with complex potentials, which will be crucial to the understanding of energy localization in the realistic two-dimensional materials.

The cover-time problem, i.e., the time to visit every site in a system, is one of the key issues of random walks with wide applications in natural, social, and engineered systems. Addressing the full distribution of cover times for random walk on complex structures has been a long-standing challenge and has attracted persistent efforts. Usually it is assumed that the random walk is noncompact, to facilitate theoretical treatments by neglecting the correlations between visits. The known results are essentially limited to noncompact and homogeneous systems, where different sites are on an equal footing and have identical or close mean first-passage times, such as random walks on a torus. In contrast, realistic random walks are prevailingly heterogeneous with diversified mean first-passage times. Does a universal distribution still exist? Here, by considering the most general situations of noncompact random walks, we uncover a generalized rescaling relation for the cover time, exploiting the diversified mean first-passage times that have not been accounted for before. This allows us to concretely establish a universal distribution of the rescaled cover times for heterogeneous noncompact random walks, which turns out to be the Gumbel universality class that is ubiquitous for a large family of extreme value statistics. Our analysis is based on the transfer matrix framework, which is generic in that, besides heterogeneity, it is also robust against biased protocols, directed links, and self-connecting loops. The finding is corroborated with extensive numerical simulations of diverse heterogeneous noncompact random walks on both model and realistic topological structures. Our technical ingredient may be exploited for other extreme value or ergodicity problems with nonidentical distributions.

The interaction between flexural modes due to nonlinear potentials is critical to heat conductivity and mechanical vibration of two dimensional materials such as graphene. Much effort has been devoted to understanding the underlying mechanism. In this paper, we examine solely the out-of-plane flexural modes and identify their energy flow pathway during the equipartition process. In particular, the modes are grouped into four classes by their distinct symmetries. The couplings are significantly larger within a class than between classes, forming symmetry blockades. As a result, the energy first flows to the modes in the same symmetry class. Breakdown of the symmetry blockade, i.e., inter-class energy flow, starts when the displacement profile becomes complex and the inter-class couplings bear nonneglectable values. The equipartition time follows the stretched exponential law and survives in the thermodynamic limit. These results bring fundamental understandings to the Fermi-Pasta-Ulam problem in two dimensional systems with complex potentials and reveal clearly the physical picture of dynamical interactions between the flexural modes, which will be crucial to the understanding of their contribution in high thermal conductivity and mechanism of energy dissipation that may intrinsically limit the quality factor of the resonator.

Chaotic time series exhibit intrinsic multi-scale correlations that make long-term forecasting extremely challenging. In this paper, we develop a hybrid parallel architecture that integrates transformer and bidirectional long short-term memory (BiLSTM) networks, designed to capture complementary aspects of chaotic dynamics. The transformer branch extracts global dependencies arising from long-range temporal correlations, while the BiLSTM branch resolves short-scale structures linked to local stretching and folding mechanisms. Their jointly learned representations are fused through a feature-integration module, yielding a unified description that enhances predictive accuracy and robustness. We evaluate the framework on the Lorenz system across two representative tasks: (i) autonomous evolution prediction using time-delay embeddings and (ii) inference of unmeasured variables, a fundamental problem in nonlinear observability. The hybrid model consistently outperforms single-branch architectures in both tasks. These results highlight the utility of combining global attention mechanisms with local recurrent dynamics for modeling complex systems. Remarkably, in the inference task, it achieves stable reconstruction over arbitrarily long time horizons, which provides a practical tool for monitoring variables that are difficult or hazardous to measure, with implications for engineering diagnostics, environmental and physiological monitoring, and the analysis and control of high-dimensional nonlinear dynamics.

On-site potentials are ubiquitous in physical systems and strongly influence their heat transport and energy localization. These potentials will inevitably affect the dynamical properties of q breathers (QBs), defined as periodic orbits exponentially localized in normal mode space. By integrating on-site terms into the Fermi-Pasta-Ulam-Tsingou-β system, this work utilizes numerical simulations and Floquet analysis to systematically explore the influence of on-site potentials on QB stability. For most QBs, except those at the phonon band edges, the instability is primarily governed by parametric resonance and effectively described by coupled Mathieu equations. This approach provides a theoretical expression for the instability thresholds, which aligns well with numerical results. The instability threshold is highly sensitive to the seed mode, in stark contrast to systems without on-site potentials. In addition, a systematic stability analysis across three-dimensional parameter space shows that quartic intersite nonlinearity monotonically enhances QB instability, while quartic on-site nonlinearity yields nonmonotonic effects, in particular, an initial destabilization followed by restabilization at high amplitudes. The quadratic on-site potential uniformly promotes stability. Furthermore, these instability phase diagrams highlight both the individual contributions and combined effects of each possible component in regulating the QB dynamics. These findings offer valuable insights into QB stability and the manipulation of localized excitations in diverse physical systems with on-site potentials.

The Collatz map is investigated from a nonlinear-dynamics perspective with emphasis on the structure of its iterative orbits. By embedding integers within Sharkovsky’s ordering, odd initial values are shown to be sufficient for a complete characterization of dynamics. A “direction-phase” decomposition is introduced to separate iterative orbits into upward and downward phases, yielding a family of recursive functions parameterized by the number of upward phases. This formulation reveals a logarithmic scaling relation between the total iteration count and the initial value, confirming finite-time convergence to the period-three orbit. The Collatz dynamics is further shown to be dynamically equivalent to a binary shift map, whose ergodicity implies inevitable evolution toward attractors, thereby reinforcing convergence. Numerical analysis indicates that attraction basins follow a power-law distribution and display pronounced self-similarity. Moreover, odd integers grouped by upward-phase counts are found to follow Gamma statistics. Beyond its research implications, the framework provides a concise pedagogical case study illustrating how nonlinear dynamics, symbolic dynamics, and statistical characterization can be integrated to analyze a classical discrete problem.